Re: Misinterpretation of the radial parameter in the Schwarzschild solution?

LEJ Brouwer says...

I am writing 't' as the timelike coordinate and 'r' as a spacelike
coordinate. You are saying that 'r' is not a coordinate at all in the
interior, but just a parameter,

I said the *radius* is a parameter. r is *not* the radius of anything
in the interior.

which would seem to kill any attempt to interpret the interior
region as part of physical spacetime.

Why do you say that?

Let's look at the Schwarzchild metric again in full gory detail:

ds^2 = (1-2m/r) dt^2 - 1/(1-2m/r) dr^2
- r^2 dTheta^2
- r^2 sin^2(Theta) dPhi^2

In the neighborhood of some point with coordinates
t0, r0, theta0, phi0 (with r0 < 2m), we can use local
coordinates u,v,w,q with the relation to the original
coordinates given by:

t = t0 + 1/square-root(2m/r0 - 1) u
r = r0 + square-root(2m/r0 - 1) v
theta = theta0 + w/r0
phi = phi0 + q/(r0 sin(theta0))

In terms of these coordinates, the metric looks like this

ds^2 = (1+av) dv^2 - (1+bv) du^2 - (1+cv)dw^2 - (1+ev+fw)dq^2
+ terms that are second order in u,v,w,q

where a,b,c,e,f are all constants (you can work out what they
are in terms of r0,t0,theta0 and phi0), and I've only kept the linear
term in the expansion of the metric coefficients---that's all we
need for local coordinates.

Now, do another transformation to coordinates T,X,Y,Z
related to u,v,w,q as follows:

v = T - aT^2/4 - bX^2/4 - cY^2/4 - eZ^2/4
u = X - bXT/2
w = Y - cYT/2 + fYZ/4
q = Z - eZT/2 - fYZ/2

(I think that's right, if not, you get the idea). In terms of
X,Y,Z,T, the metric looks like this:

ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2
+ terms that are second-order in X,Y,Z and T

So in a very small region of spacetime, things look like
Minkowsky space even in the interior of the event horizon.
That's all that GR requires for "physically meaningful";
every region of spacetime locally looks like Minkowsky space.

I don't see how you can choose local coordinates in the interior
solution such that the metric takes this form. To be locally flat it
needs to have this form at least up to first order, so you can't just
replace r^2 with t^2.

See above.

--
Daryl McCullough
Ithaca, NY

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